Unobtainium is the engineering jargon for, "a material that would be perfect for our purposes, if we could get it, which we can't." Sometimes an object that actually exists, or existed at one time, becomes unobtainium because it's unavailable now. When used in the realm of fiction, Unobtainium is usually the exotic material that is needed to make the story work. Without it, all the nifty machines and plot-enabling gadgets quit functioning. Some forms of unobtainium are based on real physics, but beyond the current scope of human engineering, such as room-temperature superconductors; they would revolutionize just about every form of technology, but they are not in and of themselves dangerous or based on some exotic physics-bending principle. Others are more fantastic "high-grade" unobtainium, such as antimatter, which would be a revolutionary way of storing huge amounts of energy, if it didn't violently (as in, 1*10^-9 grams = tactical nuclear weapon yield) undergo mutual annihilation with any conventional matter it comes into contact with, including air molecules and the walls of whatever you're trying to store the damn stuff in.
The most common varieties of unobtainium in fiction sit somewhere in the middle, like materials so resistant to heat and/or damage as to be invulnerable compared to other, similar substances. Materials such as mithril are the fantasy version. Thunderbolt Iron is especially popular in fiction (and has some basis in reality – until smelting was invented (which takes ridiculously hot - 1250 °C or 2282 °F - furnaces, hence the term "Iron Age"), it was the only source of refined iron). Much mad science uses unobtainium, such as chemicals with impossible properties, universal solvents that can dissolve anything in the blink of an eye, super-explosives that make nitroglycerin look like a weak cough, and plenty of other funny-colored solutions. Following this would be medical and/or chemical wish-fulfillers; Classical real-world alchemy casually referred to carmot, the base substance of the Philosopher's Stone, and Azoth, either the "universal medicine" or "universal solvent". The ancient Greek writer Plato referred to "Orichalcum" (Greek for "mountain bronze") in his description of Atlantis. In Science Fiction, it will usually take one of three flavors: whatever stuff makes Faster-Than-Light Travel possible, closely followed by the stuff that can mess with gravity (if they're not one and the same), and finally, the stuff they make mecha and alien spacecraft out of, which is why they tend to be effectively immune from earthly weapons or environmental damage.
The term "Unobtanium" originates from aerospace engineers in the late 1950s, where it was used as a jargon for a material sufficiently strong, light, and/or durable to meet the needs of a particular situation under discussion, even if no known material could possibly do so. It has occasionally been used in official discussions to avoid directly identifying a material whose use is still considered top secret (such as the titanium skin in the project that eventually produced the SR-71 Blackbird). Most people, however, first heard the term as the mineral sought by the mining company in Avatar, or the material used to build Virgil from The Core (And Unobtainium in both movies doesn't even make sense when you analyze it deeper. Apparently, someone consulting on the Avatar made transportation cost estimates similar to ours because supposedly a kilo of unobtainium (which is used for room-temperature superconductor) sells for $20 million--a price that makes profitability at least conceivable. But, high price usually means tiny market, so even at that price, making a profit from a mining operation on a distant moon seems tenuous at best. If the unobtainium market were indeed highly lucrative, then sooner or later someone would figure out how to synthesize it. Surely a culture that can go to a moon, and at that a moon in a different star system, could figure out how to make unobtainium. As for The Core, it does not explain the material further other than 'get stronger and generates power in the presence of extreme heat'. But the 'generate power' part is what makes it ridiculous: At the end of the movie, after removing the reactor's fuel rods, the remaining terranauts are stranded with no power source, only 12 minutes from certain death due to the blast wave, but don't worry: The unobtainium hull of the ship conveniently converts heat to energy. In the remaining 12 minutes they pull the ship's power wires loose and solder (what!?) them to the ship's inner hull. The ship magically powers up and the terranauts surf the blast wave to safety. This left us with many questions. First where's the circuit? To produce power, current has to flow and this requires a voltage drop. If one wire is hooked to the ship's hull which supposedly acts as a high-voltage source, then where is the ground wire connected? Even if the circuit did exist, how could the unobtainium possibly produce exactly the right kind of power (AC verses DC) at the correct voltage just by dumb luck? If it's so easily done then why wasn't the ship designed with unobtainium backup power in the first place?)
There are many real world examples of unobtainium. While the ideas of Mithril and Vibranium actually existing on our earth may be laughable, the idea of a mineral/resource that is near impossible to obtain is almost historical. Rare-earth elements are used in most modern electronics, and aren't really rare, but they are hard to find in an economically-usable state. And, in addition, 97% of rare-earth mining is done in China. Because of their usefulness, worries that the Chinese could cut off or severely reduce exports of it is enough that now others countries are looking into reopening mines almost solely so that the Chinese cannot make unobtainium of them. Some of the example:
Aluminium: When Aluminium was first discovered, it was considered unobtainium, because although it's actually the most common metallic element on Earth, it was very difficult to extract from its ores. Hence, The Washington Monument was capped with a pyramidal ingot of pure aluminum, Napoleon III's sets of dinnerware made from aluminum, and the statue of Anteros in London. Then Hall (and, independently, Héroult) discovered an easy way to make aluminium, and it stopped being unobtainium two years later with the construction of the first large-scale aluminum refining plant. You're probably drinking out of an aluminum can right now. Even today, separating aluminum isn't as easy as you might think — it requires huge amounts of electrical power. Raw ore (bauxite) gets shipped around the world to take advantage of the cheapest possible electricity prices (Iceland smelts bauxite from South America to take advantage of its surplus geothermal power, while Quebec does the same to take advantage of its abundant hydroelectric power). It's also heavily recycled, as melting it down and re-casting it only requires 5% as much power as refining it.
Astatine: it was estimated that the amount of astatine in the planet barely can be gathered in a spoon, with around 30 grams existing on the entire Earth at any one time. This is because it is a product of radioactive decay, but is radioactive itself, with a half-life of 8.3 hours before decaying to lead. It is also so radioactive that if you had enough astatine to be able to see it with the naked eye, you'd be dead from radiation poisoning in minutes.
Titanium: During the Cold War, most of the significant titanium mines were either in the Soviet Union or elsewhere in the Eastern Bloc. As a result, Western aircraft designers often half-jokingly referred to the stuff as "unobtainium." Eventually, new mines were discovered in Australia, South Africa, Canada, and Norway—all safely outside Soviet influence—and titanium stopped being unobtainium for the West. U.S. aircraft designers during this period are the one who popularize the idea. On the other hand the largest producer in the world is still in Russia. Though they now sell to everyone, and actually almost all titanium Boeing now uses is bought from them.
Antimatter: During the Cold War, the US Air Force had a strong desire to develop antimatter bombs, perhaps feeling that hydrogen bombs just weren't apocalyptic enough. Fortunately, there is no known natural source of antimatter and no practical way to make it that can produce the macroscopic quantities of the stuff needed for bombs, and no practical way to contain the stuff safely enough for long enough to make such a weapon useful - critically, a nuclear bomb will not explode unless you want it to, while an antimatter bomb will always try to explode whether you want it to or not.
Weapon-grade nuclear material: Fortunately for those concerned about rogue states or terrorists developing nuclear weapons, the required fissionable material is this. Plutonium does not occur naturally and must be manufactured in a nuclear reactor designed for the purpose. Uranium, while it can be mined, is all but useless for fueling a nuclear reaction in its natural state and must be enriched (U-235, the isotope of uranium used in both reactors and bombs, comprises less than 1% of natural uranium; reactor fuel contains over 30% U-235 and uranium bombs over 90%). The techniques involved in uranium enrichment require vast amounts of resources and specialized equipment, making it an expensive prospect for even a nation to attempt, let alone a terrorist group (The plant that processed the uranium for the Hiroshima bomb was housed in a building over 1 mile in length and took up over 1.6 million square feet of floor space; more modern techniques involve equipment that is both sufficiently specialized that sales are highly regulated and sufficiently complex that manufacturing them requires only slightly less unobtainium-esque materials)
Wootz steel: This is a very specific type of steel. It's made out of crucible-fired sand consisting of iron and tungsten carbide, which only naturally occurs in a very few places, almost all of them in central Asia. The process for making it was lost for centuries after the ore ran out, and was only rediscovered very recently through chemical analysis (the ore contained trace amounts of vanadium that created an unusual spiky crystal structure in the solidifying ingots). By all accounts, wootz steel is both stronger and more flexible than ordinary steel; back when swords were still used as weapons, Indo-Persian swords were highly valued throughout India and the Middle East because of this.
Chlorine trifluoride: This is the evil, HAZMAT twin of unobtainium, a material of unspecified composition that greatly endangers human life with the smallest spills or leaks. Derek Lowe has a nightmarish description at his blog. From his article: "It is, of course, extremely toxic, but that's the least of the problem. It is hypergolic (combusts spontaneously) with every known fuel, and so rapidly hypergolic that no ignition delay has ever been measured. It is also hypergolic with such things as cloth, wood, and test engineers, and asbestos, sand, and water-with which it reacts explosively." There is a rumor that the Germans produced as much as 30 tons of the stuff to use as an incendiary weapon and rocket fuel during World War II, but put it into the storage after it became clear that the war is going nowhere — and then it simply disappeared somewhere... Pandemonium was in fact an early name for the transuranic metal americium, which is highly radioactive. It was called such because it was extremely difficult to separate from the element curium which was originally called delirium.
Carbon nanotubes: They have an immensely useful electronic, optical, and mechanical properties, including a strength-to-weight ratio vastly superior to any building material currently in use. Sadly, as of 2010, even poor grade nanotubes go for about $100/gram. Guess that space elevator will have to wait a few more years. The biggest problem with them at the moment is to avoid cumulative weakening, as at the moment the more nanotubes you stock together, the more the nanoscopic faults accumulate, until their strength is all but gone. Still, many scientists are confident that they'll have long and durable nanotube strings in a couple of years.
Exotic matter: Nuclear physics has created Exotic Matter in exceedingly minute quantities. Synthetic baryons (baryons are particles such as protons and neutrons) contain configurations other than the standard two up/one down, one up/two down, quark arrangements. Theoretically such femtotechnology could lead to a dazzling array of alternate chemistries. Thousands of alternate periodic tables may be possible, maybe more. The synthetic baryons all decay rapidly (the order of 10^-10 sec or shorter halflifes). The chemistry of an atom is determined by the electrons surrounding the nucleus. Atoms with synthetic baryons would be considered different isotopes of the same element. Recently a Japanese team reported the discovery of a so-called tetraquark — an exotic baryon consisting of four quarks instead of a normal three. Despite being roughly equally important to modern particle physics, the news were drowned by the buzz of a Higgs boson discovery, which has much better publicity.
Rhenium: The platinoid metal rhenium has all types of possible uses, mainly because it gives nearly magical properties to the metals it is alloyed with. Unfortunately, it is so incredibly rare and expensive that it's used mainly in aircraft engines, where its cost can be justified. (Not coincidentally, it was the very last of the stable elements to be discovered, in 1925.) There is one single concentrated deposit of rhenium on the whole planet, discharging as a sulfide gas from a single fumarole on the side of a volcano on the South Kurile island of Iturup, which is disputed between Russia and Japan. It is possible that the harder position Russia has recently taken about the Kurile issue might be explained by the wish to protect and exploit this deposit.
Wood: It was an Unobtanium for Venice, England and other similar naval empires. Not just any wood but the right kind of wood for the right jobs. Trees that had longer trunks, for instance, received favor, for their utility in building certain long parts of ships; oak was favored for many purposes—especially for warships—because it is sturdier (with live oak from the American South being particularly prized); trees with tall, straight trunks were needed to build masts; and pine was needed to produce pitch and tar, needed for waterproofing and other purposes (e.g. preventing shipworm from afflicting their vessels). This could often be an element and not always a positive one in the relations between Britain and America and Scandinavian countries, both of which were among the main suppliers. The loss of the American supply of timber, pitch, and tar for the Royal Navy was actually a key catalyst for the Industrial Revolution; the search for a source of pitch led a particularly hapless poor Scottish earl to develop processes to efficiently produce coal tar, which directly led to the first gas lighting (and therefore the 24-hour factory and shift work) and indirectly led to the modern chemical industry. Earlier the English were responsible for converting yew into the wooden Unobtanium du jour due to their legendary enthusiasm for longbows. While yew trees were common throughout Europe, good knot-free lengths suitable for bowmaking were rare. By the 15th century the English had instituted an import fee payable in yew staves on every ship coming into the country, and forests as far away as Austria were being pillaged for yew. Only the introduction of guns put a stop to the demand.